THE FUNCTIONAL INTEGRAL FOR A FREE PARTICLE ON A HALF-PLANE*
Michel Carreau
arXiv:hep-th/9208052v1 21 Aug 1992
Boston University Department of Physics 590 Commonwealth Avenue Boston, Massachusetts 02215 U.S.A. E-mail:
[email protected]
To be published in : Journal of Mathematical Physics
BU-HEP-91-23
Typeset in TEX
November 1991
* This work is supported in part by funds provided by the U. S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069, and by the Fonds pour la Formation de Chercheurs et l’Aide `a la Recherche. 0
ABSTRACT A free non-relativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Non-local boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schr¨odinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian.
1
I.
INTRODUCTION
In a paper by Clark et al., and later on by Gaveau et al. and Farhi et al. (see Ref. [1]) the functional integral for a particle moving on a half-line was derived. It was found that the measure for the functional integral depends upon the one parameter family of boundary conditions which guarantee that probability does not not leave the half-line. Farhi, Gutmann and the present author2 derived the functional integral for a free particle moving in a box in 1-D for the system subject to a four- parameter family of boundary conditions consistent with unitarity. They found that the measure for the functional integral depends upon these four parameters and that the paths are allowed to jump from wall to wall. In this paper, we show how to generalize these ideas to more than one dimension. In Section II, we describe the most general boundary condition leading to unitary time evolution for the free particle moving on a half-plane. (Equivalently, we find the self-adjoint extensions of the free Hamiltonian.) In Section III, we construct the Brownian functional integral representation of the Brownian Green’s function corresponding to each self-adjoint Hamiltonian. The Green’s function of the Schr¨odinger equation are obtained by an analytic continuation of the Brownian Green’s function. We point out how to generalize these results to more complex quantum systems. In the Appendix we give the functional integrals of a one dimensional quantum systems made out of any number of half-lines joined together at their origin. II.
PARTICLE ON A HALF-PLANE — SELF-ADJOINT EXTENSIONS
Consider the motion of a free particle on a half-plane in two dimensions (−∞ < y < +∞ and x > 0). The Hilbert space consists of all square integrable functions on the half-plane. The domain of the Hamiltonian 2 2 ˆ = −1 d − 1 d H 2 dx2 2 dy 2
(2.1)
ˆ is Hermitian, which is the condition must be chosen so that H Z +∞ Z ∞ Z +∞ Z ∞ ∗ ∗ ˆ ˆ dx dy = Hψ ψ Hφ φ dx dy −∞
−∞
0
(2.2)
0
or equivalently Z
+∞
−∞
∗ dφ dψ dy ψ ∗ − φ dx dx
=0
(2.3)
x=0
ˆ for all ψ and φ in the domain of H. ˆ The condition (2.3) can be realized most generally if for every χ in the domain of H Z +∞ dχ dy ′ g(y ′ , y)χ(0, y ′) (2.4) = (x, y) dx −∞ x=0 for each y, with g an arbitrary function obeying
g(y ′ , y) = g ∗ (y, y ′ ) . 2
(2.5)
A similar result was obtained in Ref. [3]. Let us pick the following parameterization for g(y ′ , y) o n ′ (2.6) g(y ′ , y) = [ρ(y) + α(y)] δ(y ′ − y) − ρ(y ′ )f (y ′ , y) eih(y ,y) where δ is the delta-function and α, h, ρ, f are arbitrary smooth real functions with the following restrictions: ρ, f are non-negative, h is antisymmetric in its arguments, ρ(y)f (y, y ′) = ρ(y ′ )f (y ′ , y) and Z
+∞
(2.7)
dy f (y ′ , y) = 1 .
(2.8)
−∞
We can show that g(y ′ , y) uniquely determines the functions α, h, ρ, f . Given a g satisfying ˆ to be functions obeying (2.4) then H ˆ is self-adjoint and (2.5) if we choose the domain of H probability does not leave the half-plane. We see that we have infinitely many self-adjoint extensions indexed by the function g(y ′ , y). To each self-adjoint extension there corresponds a unique Green’s function Gg (~z i ,~z f , t) with ~z a = (xa , ya ), a = i, f on the half-plane. The Green’s function can be expressed as ˆ
Gg (~z i ,~z f , t) = h~z f |e−iHt |~z i ig .
(2.9)
Thought of as a function of ~z f , Gg satisfies the Schr¨odinger equation and the boundary condition (2.4), that is Z d = dyf′ g(yf′ , yf )Gg (~z i , (0, yf′ ), t) . Gg (~z i ,~z f , t) dxf xf =0
(2.10)
If the boundary condition (2.4) has the property that when it is satisfied by χ(x, y) it is also satisfied by χ(x, y + y0 ) for an arbitrary y0 , we say it is translationally invariant in the y-direction. If we want a translationally invariant boundary condition in the y-direction, then g(y, y ′) can depend only on y − y ′ . Condition (2.4) says that the net current flowing out of the half-plane is zero. The condition that the current flow at each point on the edge of the half-plane be zero is that the integrand in (2.3) vanishes for each y. This can be realized by taking ρ(y) = 0, leading to g(y ′ , y) = α(y)δ(y ′ − y) , (2.11) the most general local boundary condition. If we want to enforce the translation invariance in the y-direction and the condition that no current flow at each y, we take α(y) independent of y. Then g is simply g(y ′ , y) = αδ(y ′ − y) . (2.12) With these restrictions there is only a one-parameter family of self-adjoint extensions. 3
III.
PARTICLE ON A HALF-PLANE — FUNCTIONAL INTEGRAL
In this section, we construct the Brownian functional integrals corresponding to the self-adjoint extensions of the free Hamiltonian constructed in the previous section. The analytic continuation of the Brownian functional integrals results in the Green’s functions of the Schr¨ odinger equation. 1.
Preliminaries
In order to have a mathematically well-defined measure on a class of paths, we work with Brownian functional integrals. On the whole plane the Brownian functional integral Z − 1 R τ ~q˙ 2wp dτ ′ 1 −|~zf −~z i |2 /2τ B e (3.1) = Awp (~z i ,~z f , τ ) = d~q wp e 2 0 2πτ
with ~q wp (0) = ~z i and ~q wp (τ ) = ~z f , is the Green’s function of the Schr¨odinger equation with imaginary time t = −iτ . We call AB wp the Brownian Green’s function. The symbol Rτ 2 ′ 1 ˙ − ~q dτ [d~q wp ] e 2 0 wp is to be interpreted as the Brownian measure on paths ~q wp . This measure gives zero weight to discontinuous paths. The Green’s function, Gwp , of the Schr¨odinger equation on the whole plane is obtained by the analytic continuation of AB wp : Gwp (~z i ,~z f , t) ≡ AB z i ,~z f , it) . wp (~
(3.2)
When working on the half-plane, we use the reflecting measure, which is induced by taking the absolute value of the x-component of whole plane Brownian motion. That is, if ~q wp = (q1wp , q2wp ) is a whole plane path then~q ∗ ≡ (|q1wp |, q2wp ) ≡ (qx∗ , qy ) lies on the half-plane and the probability distribution, or measure, on ~q wp directly induces a probability distribution on ~q ∗ . For each path ~q ∗ = (qx∗ , qy ) which goes from ~z i to ~z f on the half-plane in time τ , there exists a “local time” which tells us how much time the path spends near the boundary of the half-plane (x ≈ 0). 1 ℓ(τ ) ≡ lim ǫ→0 2ǫ
Z
0
τ
I (qx∗ (τ ′ ) < ǫ) dτ ′
(3.3)
where I(A)R = 1 or 0 according to whether A occurs or not. (More informally, we can think τ of ℓ(τ ) as 0 δ (qx∗ (τ ′ )) dτ ′ .) We now define the half-plane Brownian functional integral, Z R τ ∗2 ′ −1 ~q˙ dτ −rℓ(τ ) B Ar (~z i ,~z f , τ ) = [d~q∗ ] e 2 0 e (3.4) where r is a real parameter and the functional integral is over all half-plane reflected paths ~q ∗ = (qx∗ , qy ) from ~z i to ~z f in time τ . The [d~q∗ ] times the first exponential is to be interpreted as the reflected Brownian measure. There is a Brownian motion theorem4 that the limit in (3.3) exists for almost every path so (3.4) is sensibly defined. The Brownian motion on the half-plane, ~q ∗ (τ ′ ), is the Cartesian product of a reflecting Brownian motion in the x-direction, qx∗ (τ ′ ), and a whole line Brownian motion in the y-direction, qy (τ ′ ). Since these 4
motions are completely independent, we can write the reflecting measure on the half-plane as the product of the reflecting measure in the x-direction times the whole line measure in the y-direction. Moreover, the local time, ℓ(τ ), depends only on the motion in the x-direction. Therefore, the functional integral (3.4) factors as follows B AB z i ,~z f , τ ) = AB r (~ x,r (xi , xf , τ ) Ay (yi , yf , τ ) Z Z R τ ∗2 ′ R τ 2 ′ − 21 − 12 q˙x dτ −rℓ(τ ) q˙y dτ ∗ 0 0 ≡ [dqx ] e [dqy ] e e
(3.5)
where ~z a = (xa , ya ), for a = i, f . It follows that AB r satisfies the boundary condition d B B (3.6) = rAr (~z i ,~z f , τ ) Ar (~z i ,~z f , τ ) dxf xf =0
AB x,r
xf =0
1
since has been shown to satisfy the same boundary condition. The analytic continuation, Gr (~z i ,~z f , t) ≡ AB z i ,~z f , it) is the Green’s function of the Schr¨odinger equation r (~ for a free particle on the half-plane satisfying (3.6). This one-parameter family of Green’s functions corresponds to the one-parameter family of self-adjoint Hamiltonians defined in the last section by (2.12), i.e. those with translationally invariant local boundary conditions. 2.
General Local Boundary Condition
In this subsection, we build the Brownian functional integrals for the self-adjoint extensions corresponding to the boundary condition (2.4) with g given by (2.11). Let us generalize the term, rℓ(τ ), in the functional integral (3.4) to Z τ ˙ ′ )dτ ′ α (qy (τ ′ )) ℓ(τ (3.7) 0
for a path ~q = (qx∗ , qy ) with local time ℓ, where ℓ˙ ≡ dℓ/dτ ′ and α is an arbitrary real function. The factor ℓ˙ gives a contribution different from zero whenever the paths hit the edge of the half-plane. More informally, we can think of (3.7) as Z τ Z +∞ α(y0 )δ (qy (τ ′ ) − y0 ) δ (qx∗ (τ ′ )) dy0 dτ ′ , (3.8) ∗
0
−∞
a continuous sum of delta-function potentials along the edge of the half-plane with different strengths for different y. The functional integral on the half-plane becomes Z R τ ∗2 ′ R τ ˙ ′ )dτ ′ − 21 α(qy (τ ′ ))ℓ(τ ~q˙ dτ − ∗ B 0 0 Aα (~z i ,~z f , τ ) = [d~q ] e . (3.9)
The expression (3.7) exists for almost every path5,6 so (3.9) is sensibly defined. In the next subsection we will show that the analytic continuation Gα (~z i ,~z f , t) ≡ AB z i ,~z f , it) is the α (~ Green’s function of the Schr¨odinger equation satisfying the most general local boundary condition d , (3.10) = α(yf )Gα (~z i ,~z f , t) Gα (~z i ,~z f , t) dxf xf =0 xf =0 for each yf . Let us mention that path integral representations of Green’s functions for several types of differential equations satisfying local boundary conditions have already been studied extensively in more than one dimension(see Ref. [7] and references therein). The equation (3.9) was obtained in Ref. [7] for the diffusion equation. In the next section, we generalize this result for non-local boundary conditions. 5
3.
The Functional Integral For General Boundary Condition
In this subsection, we construct a family of functional integrals AB g parameterized with the function g, introduced in Section II, which satisfies the boundary condition Z d B = dyf′ g(yf′ , yf )AB z i , (0, yf′ ), τ ) A (~z i ,~z f , τ ) g (~ dxf g xf =0
(3.11)
The analytic continuation of AB g with respect to τ results in the Green’s functions (2.9) of the Schr¨ odinger equation satisfying the non-local boundary condition (2.4) or equivalently (2.10). In order to obtain a functional integral satisfying the non-local boundary conditions (3.11), we wish to define a measure giving non-zero weight to paths that can jump to other points on the edge of the half-plane when they hit the edge of the half-plane. Moreover, this measure should depends on the four functions ρ, α, f, h, introduced in Section II to parameterized the function g, since the boundary condition (3.11) depends on them. We will see below that when a path hit the edge of the half-plane at y along y-axis, ρ(y) gives the rate at which the path jumps from that point and f (y, y ′) gives the density of probability of jumping to y ′ given it has jumped from y. The functions α(y) and h(y, y ′ ) control the size of weights and phases given to the path when it spends time near y and when it jumps from y to y ′ respectively. We will see that the probability of jumping from a point on the edge of the half-plane depends both on the time the path spends near that point and on the rate of jumping at that point. It is then natural to introduce the notion of effective local time which is roughly the amount of time the path spends near the edge of the half-plane weighted, point-by-point, by the rate at which the path jumps at that point. This notion is defined formally by (3.16) below. Let us describe the process generating the paths on the half-plane. Imagine that the path starts at ~z i ≡ (xi , yi ), with xi > 0 and undergoes ordinary reflected Brownian motion on the half-plane until its effective local time(defined below) reaches some number s1 . (At the instant the effective local time reaches s1 the path must be on the edge of the halfplane, say (0, y1−).) It then jumps to (0, y1+ ) on the edge of the half-plane. From this point it resumes its reflected Brownian motion on the half-plane until the effective local time reaches s1 + s2 at which time it reaches (0, y2−) and jumps to (0, y2+ ), and so forth. The values s1 , s2 , . . . are to be chosen as independent random variables which are exponentially distributed i.e. Prob (sk > u) = e−u . (3.12) As we will see later, the rate at which the paths jump is implicit in the definition of the effective local time, (3.16), and this is why we have set the coefficient of −u to be 1 without lost of generality. The values y1+ , y2+ , . . . are to be chosen as random variables and their distribution depends respectively on the locations (0, y1− ), (0, y2−), . . ., on the edge of the half-plane, from which the jump takes place, that is Prob w < yk+ < w + dw = f (yk−, w) dw . 6
(3.13)
We see that f (yk− , w) is the probability density of landing at (0, w) given that the path has jumped from (0, yk− ). More formally, let ~q ∗ = (qx∗ , qy ) be an ordinary reflected Brownian motion on the halfplane as described in Subsection III.1 and let ℓ be its local time defined via (3.3). Given the values s1 , y1+ , s2 , y2+ , . . . as described above, we define the path ~q by ~q(τ ′ ) ≡ (q1 (τ ′ ), q2 (τ ′ )) ≡ (qx∗ (τ ′ ), qy (τ ′ ) + ∆y1 + ∆y2 + . . . + ∆y# )
(3.14)
where ∆yk ≡ yk+ − yk− , # is the number of jump, s1 + s2 + . . . + s# ≤ s(τ ′ ) < s1 + . . . + s#+1 and ′
s(τ ) ≡
Z
τ′
(3.15)
˙ τ ) d˜ ρ (q2 (˜ τ )) ℓ(˜ τ .
(3.16)
0
Here, ρ(y) is the rate per unit local time at which the path jumps from the point (0, y) on the edge of the half-plane. We call s(τ ′ ) the effective local time. It measures the “time” the path spends near the edge of the half-plane weighted, point-by-point, by the rate ρ(y) at which the path jump at that point. Definitions (3.14) and (3.16) require further explanation since they are defined in terms of each other. Let (3.14) be given for the first #-jumps of the path, then (3.16) is well defined until s(τ ′ ) reaches s1 + s2 + . . . + s#+1 . At this time the + − ). This determines ∆y#+1 in definition (3.14), and ) and jumps to (0, y#+1 path is at (0, y#+1 (3.16) is updated to take this jump into account. s(τ ′ ) is now well defined until it reaches s1 + s2 + . . . + s#+2 ... and so forth. The above is true for # = 0, 1, 2, . . . so definitions (3.14) and (3.16) should now be clear. Note that the values y1− , y2− , . . . are completely determined by the half-plane reflected path ~q ∗ and the values s1 , y1+ , s2 , y2+ , . . .. Let τ1 , τ2 , . . ., τ# be the times at which the effective local time reaches s1 , s1 + s2 , . . . s1 + s2 + . . . + s# . We note that q2 (τ ′ ) is discontinuous at τ1 , τ2 , . . ., τ# . The positions of the path along the y-direction at time τk , before and after the k-th jump can be expressed in terms of ~q = (q1 , q2 ) as (3.17) yk± ≡ lim q2 (τk ± ǫ) . ǫ→0
− 21
Rτ
2 ~q˙ dτ ′
0 , is induced from the measure The measure on paths ~q , denoted as [d~q]ρ,f e on half-plane reflected paths and the distribution (3.12) and (3.13). We use this measure to define the functional integral
AB g
(~z i ,~z f , τ ) =
Z
− 12
[d~q]ρ,f e
Rτ 0
2 ~q˙ dτ ′
−
e
Rτ 0
˙ ′ )dτ ′ α(q2 (τ ))ℓ(τ ′
i
e
# P
k=1
h(yk− ,yk+ )
.
(3.18)
The functional integral is over all paths ~q from ~z i ≡ (xi , yi ) to ~z f ≡ (xf , yf ) in time τ . In (3.18), ρ, α, h, f are the four functions introduced in Section II in order to parameterize the function g specifying the boundary condition (2.4). # is the number of jumps of a given path ~q. The last factor in (3.18), involving h, is a product of phases where each factor is the phase associated with one jump of the path. The arguments of h represent the location 7
from and to which the path jump respectively along the y-axis. The factor involving α was discussed in the previous subsection. The memoryless property of the distribution (3.12) and the fact that the distribution + of yk , (3.13), does not depend on past events but only on the present position of the path before a jump guarantees that AB g satisfies the convolution equation Z B d2~z AB z i ,~z, τ1 ) AB z,~z f , τ2 ) (3.19) Ag (~z i ,~z f , τ1 + τ2 ) = g (~ g (~ half− plane
(To establish this, we used the fact that (3.7) and the last factor in (3.18) convolve pathby-path). AB g also satisfies ∗
AB z i ,~z f , τ ) = AB z f ,~z i , τ ) g (~ g (~
(3.20)
because for each path~q from ~z i to ~z f there exists a path~q ′ from ~z f to ~z i with weight complex conjugate to the weight of the path ~q. For ~z i and ~z f not on the edge of the half-plane, and for τ small (so that paths which hit the boundary can be neglected), AB g is well-approximated B B by Awp . Combined with (3.19) this ensures that Ag satisfies the same differential equation z i ,~z f not on the edge of the half-plane: as AB wp for all τ and for all ~ ! 1 ∂2 d ∂2 − AB z i ,~z f , τ ) = − AB (~z i ,~z f , τ ) (3.21) g (~ 2 + 2 2 ∂xf ∂yf dτ g with
AB z i ,~z f , 0) = δ (~z f −~z i ) . g (~
(3.22)
The fact that AB g satisfies the last four equations is sufficient to guarantee it satisfies some boundary condition of the form (2.4). In the next subsection, we will show that indeed AB z i ,~z f , t) ≡ g satisfies the boundary condition (3.11). The analytic continuation Gg (~ B Ag (~z i ,~z f , it) satisfies (3.11) also. So Gg is the Green’s function, (2.9), of the Schr¨odinger equation for the self-adjoint Hamiltonian corresponding to the boundary condition (2.4). We can obtain the functional integral of more complicated quantum systems as follows. For a quantum system where the particle is limited to move inside a bounded region, one can show that the self-adjoint-extension of the Hamiltonain are indexed by a function g defined on the boundary of the region. For each self-adjoint-extension, we can construct the functional integral in terms of functions ρ, f, α, h, defined on the boundary of the region, parameterizing the function g. The functional integral we get is very similar to (3.18). The reflected Brownian measure needed in the definition of the the functional integral is mathematically well-defined for region with smooth boundary and can be induced from the measure on whole line path. So far in this paper, we have only considered the motion of a free particle. We can obtain the functional integral for a particle moving under the influence of a bounded potential, U (~q), by multiplying the integrand of the functional integral (3.18) by Rτ U ~q(τ ′ ))dτ ′ − e 0 ( . (3.23) Both the self-adjoint extensions and the proof that the functional integral satisfies the appropriate boundary condition are unaffected by the introduction of that factor. 8
4.
Proof of Boundary Condition for Functional Integral In the previous subsection, we showed that the functional integral
AB g
(~z i ,~z f , τ ) =
Z
Rτ 1
−2
[d~q]ρ,f e
0
2 ~q˙ dτ ′
−
e
Rτ 0
˙ ′ )dτ ′ α(q2 (τ ′ ))ℓ(τ
i
e
# P
k=1
h(yk− ,yk+ )
(3.24)
satisfies some boundary condition of the form (2.4). In this subsection, we prove that the functional integral satisfies the specific boundary condition Z d B = dyf′ g(yf′ , yf )AB z i , (0, yf′ ), τ ) . (3.25) A (~z i ,~z f , τ ) g (~ dxf g xf =0
We note that this boundary condition, regarded as a function of ~z f , is the same for each (~z i , τ ). Also, we remark that we only need to determine AB g near the edge of the half-plane and at a particular point along the y-axis in order to distinguish which boundary condition it satisfies among the different possibilities offered by (2.4). In other words, it is sufficient to analyze AB g in the limit τ small (3.26) ~z i ≡ (xi , yi ) = (0, yi ) ~z f ≡ (xf , yf ) ≈ (0, yf ) √ where xf is much smaller than 2τ and yf = yi . As the following shows, in this limit the path integral (3.24) is well-approximated by a sum over dominant paths. By Taylor expanding the last two factors in (3.24) around the dominant paths we see that to leading orders the Brownian motion in the y-direction decouples from the jumping process and the Brownian motion in the x-direction. Once the Brownian motion in the y-direction is integrated out, the path integral we are left with is recognized to be a good approximation for the path integral of a 1-dimensional quantum system made out of a continuous set of half-lines all joined together at the origin(see the Appendix). This identification enables us to use the techniques of Ref. [2] to show which boundary condition is satisfied by the path integral. In the small τ -limit, the main contribution √ to the functional integral (3.24) comes from paths ~q(τ ′ ) that remain within a circle of radius 2τ centered at (xi , yi ) before the first jump and (xi , yk+ ) between the k-th and (k +1)-th jump. This is because between jumps the paths are generated by Brownian motion which have an exponentially small probability of reaching √ a point at a distance greater than 2τ from √ its starting point. Moreover,2,6in the small τ limit, paths for which ℓ(τ ) is greater than 2τ are exponentially suppressed . This implies, as shown in Ref. [2], that paths with #-jumps are suppressed by a factor (2τ )#/2 . One can show that paths jumping less than three times contain all of the information necessary to determine which boundary condition is satisfied by AB g . To establish this, it is sufficient to observe that, as a whole, these paths are sensible to all the values of the functions ρ, f, α, h: that is, any point on the boundary can be visited by at least one of these paths. If we had neglected paths that jump twice, this statement would not have been true. We can neglect paths that jump more than twice. 9
To sum up, we have seen that the dominant paths √ in the functional integral (3.24) jump 2τ and remain within a circle of radius less than three times, have a local time less than √ + + 2τ from the points (xi , yk ). (Note that y0 ≡ yi .) Also, we have seen that we can drop any contributions to the functional integral suppressed by at least a factor (2τ )3/2 since we have neglected paths with three jump and more. In the functional integral (3.24) the functions paths. Since a √ ρ, f, α, h depend on the + given path~q, with #-jump, deviates by at most 2τ from the points (xi , yk ), we can perform a Taylor expansion of ρ, f, α, h around these points. With a little bit of work, we can show that we need only to keep the leading terms in the expansion of these functions. For the contribution to the functional integral coming from the next to leading terms different from zero in this expansion are suppressed by a factor (2τ )3/2 . (for the case where ρ, f, α, h are very slowly varying it is clear that it is sufficient to keep only the leading term on their expansion.) By expanding ρ, f, α, h to leading order, we can rewrite the functional integral (3.24) as # #−1 P P + Z Rτ 2 ′ − α(yk+ )ℓk i h(yk+ ,yk+1 ) 1 ˙ ~ q dτ − e k=0 e k=0 (3.27) AB z i ,~z f , τ ) ≈ [d~q]ρ,f e 2 0 g (~ D
where D stands for the path integral over all √ the dominant paths from ~z i to ~z f ; that is the paths that remain within a circle of radius 2τ from the√points (xi , yk+ ), with zero, one and two jumps(# = 0, 1, 2) and with local time less than 2τ . (we have set y0+ ≡ yi and + yk− ≈ yk−1 , for k = 1, 2, . . ..) Also, 1 ℓk ≡ lim ǫ→0 2ǫ
Z
τk+1
I (q1 (τ ′ ) < ǫ) dτ ′
(3.28)
τk
is the local time the path spends near the origin between the k-th and (k + 1)-th jump for k = 1, 2, . . . , # − 1. ℓ0 and ℓ# is the local time the path spends near the origin before the first and after the last jump respectively. Finally, the effective local time in the functional integral is # X s(τ ) ≈ ρ(yk+)ℓk (3.29) k=0
to leading order in the expansion of ρ. Now, let us make two remarks. First recall that a path ~q can be expressed in terms of a half-plane reflected path ~q ∗ = (qx∗ , qy ) from ~z i to an intermediate point (xi , y˜) and in terms of the values s1 , y1+ , s2 , y2+ , . . . restricted to # X
k=1
∆yk ≡
# X
k=1
(yk+ − yk− ) = yf − y˜ .
(3.30)
This restriction ensures that the path ~q ends at ~z f as required. The path integral can be broken into a few integrations. We begin by integrating over the values s1 , y1+ , s2 , y2+ , . . . restricted to (3.30) with ~q ∗ and y˜ fixed. Next we integrate over paths qx∗ , qy with y˜ fixed and finally integrate over all y˜. As a second remark we point out that, in the functional 10
integral (3.27), by relaxing the domain of integration to be over all paths instead of only those describe by D we only introduce errors suppressed by at least a factor (2τ )3/2 . We can now rewrite (3.27) as AB g
(~z i ,~z f , τ ) ≈
Z
+∞
−∞
d˜ y AB ˜, τ ) Ayi ,yi +(yf −˜y) (xi , xf , τ ) free (yi , y
(3.31)
≈ Ayi ,yf (xi , xf , τ ) where AB ˜, τ ) = √ free (yi , y
2 1 e−(˜y −yi ) /2τ 2πτ
(3.32)
is the result R τ 2 of′ the integration over all the paths qy (note that to leading order nothing 1 − ~q˙ dτ but e 2 0 depends on ~q y in (3.27).), and Ayi ,yf is the result of the integration over the values s1 , y1+ , s2 , y2+ , . . . restricted to (3.30) and over all paths qx∗ . In the appendix we show that Ayi ,yf can be thought of as the functional integral of a 1-dimensional system made out of a continuous set of half-lines all joined together at the origin. This system is a straightforward generalization of a quantum system with only two half-lines connected at the origin and this system was studied in a previous paper2 . Using the same techniques introduced in Ref. [2] we can show that Ayi ,yf satisfies Z +∞ d dyf′ g(yf′ , yf )Ayi ,yf′ (xi , 0) = Ay ,y (xi , xf ) dxf i f −∞ xf =0
(3.33)
and in turn that implies that AB g satisfies (3.25) as we wanted to show. IV.
CONCLUSIONS
In this paper, the equivalence between the operator and functional approaches to quantum mechanics established for one-dimensional quantum systems in Ref. [1,2,3] has been extended to quantum systems in higher dimension admitting general boundary conditions. For each self-adjoint extension the Green’s function is obtained as the analytic continuation of the functional integral whose measure depends explicitly upon the parameters defining the extension. ACKNOWLEDGMENTS I am grateful to Edward Farhi and to Sam Gutmann for their supports, insights and valuable remarks throughout the course of this work. I wish to thank Daniel Stroock, Barrett Rogers, Martin Leblanc and the referee for helpful discussions and comments.
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REFERENCES 1. T.E. Clark, R. Menikoff and D. H. Sharpe, Phys. Rev. D22 (1980) 3012. E. Farhi and S. Gutmann, Int. Jour. Mod. Phys. A5 (1990) 3029. B. Gaveau and L. S. Shulman, J. Phys. A.: Math. Gen. 19 (1986) 1833. 2. M. Carreau, E. Farhi, S. Gutmann, Phys. Rev. D42 (1990) 1194. 3. M. Carreau, E. Farhi, S. Gutmann and P. Mende, Ann. of Phys. 1,(1990) 186. 4. K. Itˆo and H. McKean, Jr., Diffusion Processes and Their Sample Paths (SpringerVerlag, New York, 1964), p. 63. 5. Daniel Revuz, Marc Yor, Continuous Martingales and Brownian Motion (SpringerVerlag, New York, 1991) Chap. VI. 6. K. Sato and H. Tanaka, Proc. Japan Acad. 38 (1962) 699 7. K. Sato and T. Ueno, J. Math. Kyoto Univ. 4 (1965) 529 N. Ikeda, Mem. Coll. Sci. Univ. Kyoto, Ser. A, 33 (1961) 367
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APPENDIX Consider the quantum system made out of N-half-lines parameterized by k = 1, 2, . . . , N all joined together at the origin as indicated in Fig.1. We locate the position of a particle on this system by specifying the branch, k, where the particle moves, and its distance, x, from the origin. The wave function χk (x) for each branch k of this system satisfies the Schr¨odinger equation. In order to make the Hamiltonian self-adjoint the wave function must satisfy the boundary condition N X d = Mk,j χk (0) χj (x) dx x=0
(A.1)
k=1
where M is a given hermitian matrix, characterizing a particular self-adjoint extension of this system. We can parameterize M in terms of the real parameters ρj , αj , fj , hj as follows Mk,j = (ρj + αj )δk,j − ρk fk,j eihk,j .
(A.2)
where ρk ≥ 0, fk,j ≥ 0, fk,k = 0, hk,j = −hj,k , ρk fk,j = ρj fj,k and N X
fk,j = 1 .
(A.3)
j=1
Below we will use these functions to construct the functional integral. Let us describe the process generating the paths on this system. The path starts on branch i at xi and undergoes reflected Brownian motion on this branch until its effective local time(defined below) reaches s1 . At this time the path is at the origin and it jumps to branch i′ at its origin and the path resumes its reflected Brownian motion until its effective local time reaches s2 and so forth. The values s1 , s2 , . . . are chosen to be independent random variables which are exponentially distributed, i.e. Prob (sk > u) = e−u .
(A.4)
Let k0 , k1 , . . ., k# be the branch in which the path is moving before the first jump, after the first, second and last jump respectively. We choose the values k1 , k2 , . . . to be random variables with the following distribution (note that k0 ≡ i is the branch in which the path starts its motion.) Given that the path has jumped from branch kj , the distribution of the value kj+1 is Prob (kj+1 = k ′ ) = fkj ,k′ . (A.5) We see that fkj ,k′ is the probability of jumping to branch k ′ given that the path had jumped from branch kj . Let ℓ0 , ℓ1 , . . ., ℓ# be the local time the path spends near the origin in the branch k0 before the first jump, and on branch k1 , k2 , . . ., k# after the first, second, ..., last jump respectively, we define the effective local time as s(τ ′ ) =
# X j=0
13
ρkj ℓj (τ ′ )
(A.6)
where # is the number of jumps of a given path and ρkj is the rate per unit local time at which the paths jump from the origin of branch kj . More formally let qx∗ be a 1-dimensional reflected Brownian motion on the positive half-line. With s1 , k1 , s2 , k2 , . . . as described above, we define the path q¯ by ′
q¯(τ ) =
qx∗ (τ ′ )
; in branch kj
(A.7)
where s1 + s2 + . . . + sj ≤ s(τ ′ ) < s1 + . . . + sj+1 .
(A.8)
The distribution (A.4) and (A.5) and the reflected Brownian measure for each path qx∗ induces a measure for each path q¯. The functional integral for this system is
AB i,j
(xi , xf , τ ) =
Z
Rτ 1
−2
[d¯ q]ρ,f e
0
˙2
q¯ dτ
′
−
e
# P
j=0
#−1
αkj ℓj i
e
P
j=0
h(kj ,kj+1 )
(A.9)
where the path integral is over all paths q¯ from xi on branch i to xf on branch j in time τ . Note that k0 ≡ i, k# ≡ j and that eihk,k′ is a phase given to the path whenever it jumps from branch k to k ′ . Equivalently the path integral can be perform by first integrating over all values s1 , k1 , s2 , k2 , . . . restricted to # X (kj − kj−1 ) mod N = j − i (A.10) j=1
with qx∗ fixed and then integrating over all paths qx∗ . The functional integral (A.9) satisfies N X d B = Mk,j AB Ai,j (xi , xf ) i,k (xi , 0) dxf xf =0
(A.11)
k=1
where M is given by (A.2). The proof of (A.11) is a straight forward generalization of the proof of the corresponding result for N = 2 given in Ref. [1] and will not be repeated here: for N = 2, (A.11) reduces to the Eq.(3.22) of Ref. [1] since hermiticity of the matrix M implies that ρ1 = ρ2 , f1,2 = f2,1 = 1 and h1,2 ≡ θ. Now consider the quantum system made out of an infinite number of half-line parameterized by −∞ < y < ∞ all joined together at the origin. The path on this system starts at xi on branch y0 jumps to branch y1 , y2 , . . ., y# whenever its effective local time reaches s1 , s2 , . . ., s# respectively. We denote as ℓ0 , ℓ1 , . . ., ℓ# the local time the path spends near the origin before the first jump, after the second, third, ..., last jump respectively. The effective local time is defined as # X ′ ρ(yk )ℓk (τ ′ ) (A.12) s(τ ) = k=0
14
where # is the number of jumps of a given path. The values s1 , s2 , . . . are distributed as in (A.4). The values y1 , y2 , . . . are distributed as follows. Given that the path has jumped from the origin of branch yk , the distribution of the value yk+1 is Prob (ν < yk+1 < ν + dν) = f (yk , ν) dν .
(A.13)
The paths q¯ are defined as in (A.7) with kj replace by yj . The functional integral for this system is
AB yi ,yf (xi , xf , τ ) =
Z
Rτ 1
−2
[d¯ q ]ρ,f e
0
˙2
q¯ dτ
′
−
e
# P
#−1
α(yk )ℓk i
k=0
e
P
h(yk ,yk+1 )
k=0
(A.14)
where the path integral is over all paths q¯ from xi on branch yi to xf on branch yf in time τ . Note that y0 ≡ yi , y# ≡ yf and the function f , ρ, h and α are the same functions introduced in section II. Equivalently the path integral can be perform by first integrating over all values s1 , y1 , s2 , y2 , . . . restricted to # X
(yk − yk−1 ) = yf − yi
(A.15)
k=1
with qx∗ fixed and then integrating over all paths qx∗ . We can see now that the path integral Ayi ,yf in (3.31) is given by the path integral (A.14). We can show that the functional integral (A.14) satisfies Z +∞ d B = dyf′ g(yf′ , yf )AB A (xi , xf ) yi ,yf′ (xi , 0) dxf yi ,yf −∞ xf =0
which is the generalization of (A.11).
15
(A.16)
Figure Captions Fig. 1: Quantum system made out of N-half-lines joined together at the origin. As an example of a particle trajectory, a particle can start at xi on branch 1 and can end up at xf on branch 3.
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